In a rhombus MPKN with an obtuse angle K, the diagonals intersect each other at point E. The measure of one of the angles of a triangle PKE is 16 degrees. Find the measure of the other angles of this triangle as well as the measures of the angles of triangle PMN.
Since MPKN is a rhombus, all the angles are equal. Let's denote the angle PKE as x degrees. Since triangle PKE is a triangle, the sum of its angles is 180 degrees. Therefore, we have:
x + x + 16 = 180
2x + 16 = 180
2x = 180 - 16
2x = 164
x = 82
So the measure of the angle PKE is 82 degrees.
Since MPKN is a rhombus, PKN is also a rhombus. This means that PMN is also a triangle, and its angles must add up to 180 degrees.
Let's denote the angles in triangle PMN as P, M, and N. Since KN is a diagonal of the rhombus, angle K is equal to 180 - 82 = 98 degrees.
Since MPKN is a rhombus, angle P = angle K = 98 degrees.
Since MPKN is a rhombus, the opposite angles are equal. Therefore, angle M = angle N = (180 - 98)/2 = 41 degrees.
So, in triangle PKN, the measure of the angles are: P = 98 degrees, K = 82 degrees, and N = M = 41 degrees.
In triangle PMN, the measure of the angles are: P = 98 degrees, M = N = 41 degrees.
Therefore, the measure of the other angles of triangle PKE are: K = 82 degrees, and E = 180 - 82 - 16 = 82 degrees.
In triangle PMN, the measure of the angles are: P = 98 degrees, M = N = 41 degrees.